Intersections of essential minimal prime ideals. (English) Zbl 1313.13006

Taherifar, A.

Commentat. Math. Univ. Carol. 55, No. 1, 121-130 (2014).

Summary: Let \(\mathcal {Z(R)}\) be the set of zero divisor elements of a commutative ring \(R\) with identity and \(\mathcal {M}\) be the space of minimal prime ideals of \(R\) with Zariski topology. An ideal \(I\) of \(R\) is called strongly dense ideal or briefly \(sd\)-ideal if \(I\subseteq \mathcal {Z(R)}\) and \(I\) is contained in no minimal prime ideal. We denote by \(R_{K}(\mathcal {M})\), the set of all \(a\in R\) for which \(\overline {D(a)}=\overline {\mathcal {M}\setminus V(a)}\) is compact. We show that \(R\) has property \((A)\) and \(\mathcal {M}\) is compact if and only if \(R\) has no \(sd\)-ideal. It is proved that \(R_{K}(\mathcal {M})\) is an essential ideal (resp., \(sd\)-ideal) if and only if \(\mathcal {M}\) is an almost locally compact (resp., \(\mathcal {M}\) is a locally compact non-compact) space. The intersection of essential minimal prime ideals of a reduced ring \(R\) need not be an essential ideal. We find an equivalent condition for which any (resp., any countable) intersection of essential minimal prime ideals of a reduced ring \(R\) is an essential ideal. Also it is proved that the intersection of essential minimal prime ideals of \(C(X)\) is equal to the socle of \(C(X)\) (i.e., \(C_{F}(X)=O^{\beta X\setminus I(X)}\)). Finally, we show that a topological space \(X\) is pseudo-discrete if and only if \(I(X)=X_{L}\) and \(C_{K}(X)\) is a pure ideal.

MSC:

13A15	Ideals and multiplicative ideal theory in commutative rings
54C40	Algebraic properties of function spaces in general topology

Keywords:

essential ideals; \(sd\)-ideal; almost locally compact space; nowhere dense; Zariski topology